You are asking how to get to the end of the game from an area you can access at the beginning of the game...cute.

Just leave LA from the South gate, entering Bloodtide Coast. Go South (don't exit to the East) and go into Sparkfly Fen. Continue South (again, don't exit East) into the Straits of Devastation. After you pass Ft. Trinity go South a little more then head West. Continue across the broken terrain (no clear roads) and fight your way til you reach the West exit, which is about 3/4 or so down on the zone (Sentry Waypoint).

You'll see a tank, and go through the portal behind it and enter Malchor's Leap.

There is a rough road that will take you through the Eastern Collonade (it's fallen or under attack over 50% of the time) and head West until you finally reach the Karst Waypoint.
From there you can continue South into the Cursed Shore if you want. That's the end of the road.

It's a pretty long trip from LA, but I'll try a description. First, go to the Bloodcoast Ward waypoint at the south of LA and exit into Bloodtide Coast. Head directly east along the road until you enter Lornar's Pass. Continue east until you hit the Rocklair PoI, which is on the main north/south road. Follow the main road south all the way. It will take an eastward bend after you get through the mountains at Guutra's Homestead waypoint, but stick with it all the way to the Timberline Falls portal.

Continue going as directly south as you can along the western side of Timberline Falls. The road isn't continuous, but follow it as best you can for fastest travel, passing near or through a large asura lab, a Pact fortress, the Ruins of Rankor, and across the swamp to the Talus waypoint. Go through the portal to Mount Maelstrom.

In Mount Maelstrom, head west for a bit until you pick up a road heading southwest. This road splits at a pass further down, with one branch turning back northwest. Continue southwest, towards the Firebreak Fort waypoint, and keep going until you're just past the "Help Pixar" renown point. There will be another split in the road, with a branch heading directly west across a bridge over the swamp. Take that branch and you will soon reach the Gloomirk Outpost PoI, where you will take the portal into Straits of Devastation right outside Fort Trinity.

From here, there are two reasonable ways to go.

1. Go south in the Straits and follow the coast until you reach the bridge. Cross the bridge and continue west across the archipelago to Lone Post. From here you will have to head a bit south of west until you get to the Lasciate Gateway PoI. You will have to fight a veteran gate guard to get the gate open to proceed. Then head northwest to the Rally waypoint.

2. Enter Fort Trinity and proceed west to the bay. Jump in and swim southwest to the Conquest Marina waypoint. You will then need to make your way a short distance west across broken territory to the Rally waypoint. You will have to go out of your way quite a bit to the north to get around the hills, though, and then back south.

From the Rally waypoint, head southwest across Triumph Plaza to the Sentry waypoint. Portal into Malchor's Leap.

In Malchor's Leap, you want to head for the Colonnade waypoint next, which is directly west. However, you may find it easier to head south, then west, then north in a "U" shape through Theater of Delight to get there. (Malchor's Leap has even more "can't get there from here" than Straits of Desolation.) Cross the bridge west and keep to as westerly a course as you can until you pass the Wren waypoint to your right. Then head southwest to the Karst Plains.

Good luck!

Edit: Ninja'd by Chuckles. I think my route may be a bit easier to travel than going through Bloodtide and Sparkfly, even though it's less direct. You have more clear roads to travel on in my route to Fort Trinity, which will probably get you in fewer fights along the way. And you're less likely to get lost on the roads than while finding your way across the trackless swamps and bodies of water on the direct route.

I've not been to some of the areas; and saw for the armor suggestions to use karma to buy some at the Temple of Melandru; and all the web articles talked about getting there from Karst Plains as a start. Nothing shared about how to get to Karst Plains.